Deductive Structure and Statements of Logic Lesson 1.7/1.8 Deductive Structure and Statements of Logic Objective: Recognize that geometry is based on deductive structure, identify undefined terms, postulates, and definitions, understand the characteristics of theorems, recognize conditional statements, recognize the negation of a statement, the converse, inverse, and contrapositive, and draw conclusions using the chain rule.

Definitions Def. Deductive Structure is a system of thought in which conclusions are justified by means of previously assumed or proved statements. Note: every deductive structure contains 4 elements: Undefined terms Assumptions known as postulates Definitions Theorems and other conclusions

Definitions Def. A Postulate is an unproven assumption (In other words, it is so obvious, it does not need to be proved) Def. A Definition states the meaning of a term or idea. Note: Definitions are reversible! Example: Original Definition: Reversed Definition:

Conditional Statements All definitions are stated in a specific form: “If p, then q” This type of sentence is called a Conditional Statement (or an Implication) The “if” part = the hypothesis The “then” part = the conclusion We write this mathematically as: .

Conditional Statement Example: Write the following statement in its conditional form: “Two straight angles are congruent”

To write the converse of a statement, you reverse parts p and q. is: To write the converse of a statement, you reverse parts p and q. Important Note! Because definitions can be reversed, the conditional statement (the original) and the converse will always be true. This is not always the case for theorems and postulates!

Converse Example: Conditional Statement: “If it is raining, then worms come out.” Converse: If worms come out, then it is raining

Negation The negation of any statement p is the statement “not p.” The symbol for “not p” is “~p” Ex. If p = It is raining then ~p = _____________

Converse, Inverse, and Contrapositive Every Conditional statement , has 3 other statements: Converse: Inverse: Contrapositive:

The AZ Example: Conditional Statement: Write each form of the conditional and decide whether the statement is true or false. Conditional Statement: “If you live in Phoenix, then you live in AZ.” Converse: If you live in AZ, then you live in Phoenix. Inverse: If you do not live in Phoenix, then you do not live in AZ. Contrapositive: If you do not live in AZ, then you do not live in Phoenix

Try making Venn Diagrams for each example written on the last slide. Theorem 3 If a conditional statement is true, then the contrapositive of the statement is also true. Note: Often times mini Venn Diagrams are useful in determining whether or not a conditional statement and its converse, inverse, or contrapositive are logically equivalent. Try making Venn Diagrams for each example written on the last slide.

Chains of Reasoning Many proofs we do involve a series of steps that follow a logical form. Often times it looks something like this: This is called the chain rule, and a series of conditional statements is known as a chain of reasoning. Example: If you study hard, then you will earn a good grade, and if you earn a good grade, then your family will be happy. We can conclude: If you study hard, your family will be happy. 

Draw a conclusion from the following statements: Example Draw a conclusion from the following statements: If gremlins grow grapes, then elves eat earthworms. If trolls don’t tell tales, then wizards weave willows. If trolls tell tales, then elves don’t eat earthworms. Hint: Rewrite these statements using symbols, then rearrange the statements and use contrapositives to match the symbols!

Homework Lesson 1.7/1.8 Worksheet